Most old Space Opera follows closely thereafter, including Star Wars (with it's old school dogfighting and Jedi powers), about equally as soft as Star Trek (especially when you consider the whole Trek series as a single continuity - some individual episodes are harder). These are shows that just don't worry about getting the science right, as long as it looks cool and carries the plot.

Next you get shows with Imported Alien Phlebotinum, where there's some sort of poorly understood technology that breaks the laws of physics. In some cases this ends up as green rocks that break physics in different ways in different episodes.

Then we get shows that use Minovsky Particles or the like, where they actually explain how and why the physics is broken, and try real hard to justify all departures from reality.

Then there's the shows that just have FTL - aside from some Faster Than Light technology, they stick to physics.

The scale gets a little wonky at that point, because if follows the FTL section with some things that are overall slightly softer, but they get Relativity right and don't have FTL, so they get a pass. I guess. Seems cheap to me.

After that comes Unobtanium, which is to say, shows that have theoretical particles, or advanced technology, but manage to make it all seem plausible. Definitely no FTL.

The last category is Next Sunday AD, settings so realistic they could happen, and soon.

The soft end of the scale is much easier for the GM to run, 'cause you never have to fact-check. The difficulty, however, may rest with the players. It's hard to deduce the correct solution to problems when the laws of physics are mutable. In a harder game you know to stay away from black holes, but in a soft science game, the GM might be expecting you to dive in and visit the mirror universe.

The hardness of the setting sets what types of characters are playable.

In a really hard setting, the career astronaut is a viable character type, and defines who you are. The mad scientist just gets laughed at. Alien races are few and far between.

In a really soft setting, being an astronaut is just one thing on your resume, and everyone is assumed to be comfortable in space. Mad scientists are common villains, or possibly a playable character class. There are Loads and Loads of Races.

It's all about expectations. Even though the laws of physics are not reliable in Star Trek, it's still typically functional as a Campaign Setting. You don't have scientific laws to rely on, and you can't just google search some negative space wedgie works. However, you do have an understanding of the sort of things that might work within the setting. Fire them off shot-gun style, and see which solution sticks. That's often what the characters on the show do.

Which suggests that published settings can get away with much looser and softer science than your homebrew setting can, because the players can take cues from episodes and movies.

RPGs encourage us to think in-character, and take things seriously. Players often do stop to think about the sort of logic holes that you might ignore in a movie, so you'll want to at least be prepared with some answers to the likely questions. Meaning that your preferred level of hardness may or may not work in gaming, depending on how you GM.

If you keep the plot rolling and try to stay self-consistent, it may work just well enough to suspend disbelief.

Especially if the PCs are not scientists. A space marine doesn't worry about how to solve some quantum paradox, he just shoots it. If shooting doesn't work, he retreats and nukes it from orbit. If that fails he retreats further and asks HQ to send in the eggheads.

However, that may limit your ability to have scenarios where the players have to solve a puzzle, because there's no functional laws of physics for the players to understand and apply. As mentioned in regards to Star Trek, if you didn't have the tropes and examples from the show to guide your actions, you'd be lost.

You may hand wave past this hurdle, by making Reverse the Polarity a standard action with reasonably intuitive results, or by just abstracting the solutions down to die rolls and resource allocation. In the later paradigm, you're basically saying the players don't need to know how their characters saved the day, which may or may not work for some gamers.